Solving for a Variable: Practice

Undo the operations in reverse: subtract the added $b$ first, then divide by the coefficient $a$.

$ax = c - b, \qquad x = \frac{c - b}{a}$

Dividing before subtracting would give the wrong $\dfrac{c}{a} - b$.

Subtract $2w$ from both sides, then divide the whole result by $2$.

$2l = P - 2w, \qquad l = \frac{P - 2w}{2}$

Subtract $b$ from both sides, then divide the whole numerator by $m$.

$mx = y - b, \qquad x = \frac{y - b}{m}$

Multiply both sides by $2$ to clear the fraction, then divide by the base $b$.

$2A = bh, \qquad h = \frac{2A}{b}$

Multiply both sides by $2$, then divide by the height $h$.

$2A = bh, \qquad b = \frac{2A}{h}$

Subtract $2x$ from both sides to isolate $y$.

$y = 6 - 2x$

The term $2x$ crosses over with the opposite sign, and its coefficient $2$ stays attached.

Subtract $3x$ from both sides, then multiply by $-1$ so that $y$ is positive.

$-y = 5 - 3x, \qquad y = 3x - 5$

Multiplying by $-1$ flips the sign of every term.

Multiply both sides by the reciprocal $\frac{9}{5}$ to free the parenthesis, then add $32$.

$\frac{9}{5}C = F - 32, \qquad F = \frac{9}{5}C + 32$

Add $b$ to both sides, then divide by the coefficient $a$.

$ax = c + b, \qquad x = \frac{c + b}{a}$

Subtract $ax$ from both sides, then divide the whole numerator by $b$.

$by = c - ax, \qquad y = \frac{c - ax}{b}$

Multiply both sides by $x$ to clear the denominator, then divide by $m$.

$mx = y, \qquad x = \frac{y}{m}$

The last step needs $m \neq 0$, and $x = 0$ is excluded either way, since it would make the original denominator zero.

Multiply both sides by $2$ to clear the denominator, then subtract $b$.

$a + b = 2Q, \qquad a = 2Q - b$